AMC 8 · 2014 · #21

Grade 6 number-theory

Archetype Small-Domain Constrained Search

divisibility-rulesdigit-summodular-arithmetic digit-constraintsmodular-arithmeticsystematic-enumeration ↑ Prerequisites: divisibility-rulesmodular-arithmetic

📏 Medium solution 💡 3 insights

Problem

The $7$ -digit numbers $\underline{7} \underline{4} \underline{A} \underline{5} \underline{2} \underline{B} \underline{1}$ and $\underline{3} \underline{2} \underline{6} \underline{A} \underline{B} \underline{4} \underline{C}$ are each multiples of $3$ . Which of the following could be the value of $C$ ?

$\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }5\qquad \textbf{(E) }8$

Pick an answer.

(A)

(B)

(C)

(D)

(E)

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Toolkit + CCSS Solution

Understand

Restated: Two $7$-digit numbers, $\underline{7}\underline{4}\underline{A}\underline{5}\underline{2}\underline{B}\underline{1}$ and $\underline{3}\underline{2}\underline{6}\underline{A}\underline{B}\underline{4}\underline{C}$, are each multiples of $3$. The unknown digits $A$, $B$, $C$ are each between $0$ and $9$. Which value among the choices could $C$ be?

Givens: First number: $74A52B1$ is a multiple of $3$; Second number: $326AB4C$ is a multiple of $3$; $A$, $B$, $C$ are single digits ($0$ through $9$); Answer choices: (A) $1$, (B) $2$, (C) $3$, (D) $5$, (E) $8$

Unknowns: The digit $C$ in the second number

Understand

Plan

Primary tool: #7 Identify Subproblems

Secondary: #3 Eliminate Possibilities, #6 Guess and Check

Two unknowns ($A$, $B$) appear in BOTH numbers, but only $C$ shows up in the second one. Tool #7 (Identify Subproblems) handles this by splitting the question into two clean jobs: (1) extract what the first number's divisibility rule says about $A + B$, then (2) plug that fact into the second number's rule to pin down $C$ alone. Tool #3 (Eliminate Possibilities) is what we use at the end — scan the five answer choices and keep only the one that fits the rule we derived for $C$. Tool #6 (Guess and Check) is the natural fallback: try each answer choice with the divisibility rule and see which one passes.

Execute — Answer: A

#7 Identify Subproblems 4.OA.B.4 Step 1

Subproblem 1: turn the first number's divisibility into a fact about $A + B$.
Add the known digits of $74A52B1$: $7 + 4 + 5 + 2 + 1 = 19$.
So the full digit sum is $19 + A + B$.
For the number to be a multiple of $3$, this sum must be a multiple of $3$.
Since $19 = 18 + 1$ and $18$ is already a multiple of $3$, the leftover $1 + A + B$ must be a multiple of $3$.

$$7 + 4 + 5 + 2 + 1 = 19,\;\; 19 + A + B = 18 + (1 + A + B) \;\Rightarrow\; 1 + A + B \text{ is a multiple of } 3$$

💡 The Grade 4 "recognize multiples" idea: a sum is a multiple of $3$ exactly when the extra piece on top of a known multiple of $3$ is itself a multiple of $3$.

#7 Identify Subproblems 4.OA.B.4 Step 2

Subproblem 2: do the same for the second number.
Add the known digits of $326AB4C$: $3 + 2 + 6 + 4 = 15$.
The full digit sum is $15 + A + B + C$.
Since $15$ is already a multiple of $3$, the leftover $A + B + C$ must itself be a multiple of $3$.

$$3 + 2 + 6 + 4 = 15,\;\; 15 + A + B + C \text{ is a multiple of } 3 \;\Rightarrow\; A + B + C \text{ is a multiple of } 3$$

💡 Same divisibility rule, same Grade 4 multiple-recognition move.

#7 Identify Subproblems 6.EE.B.5 Step 3

Combine the two facts.
Step 1 says $1 + A + B$ is a multiple of $3$, so $A + B$ is one LESS than a multiple of $3$ — in other words, $A + B$ leaves a remainder of $2$ when divided by $3$.
Step 2 says $A + B + C$ is a multiple of $3$.
Subtract: $C = (A + B + C) - (A + B)$ is the difference between a multiple of $3$ and a number that leaves remainder $2$, so $C$ itself must leave remainder $1$ when divided by $3$ (because $0 - 2 = -2 \equiv 1$ on a $0,1,2$ cycle).

$$A + B \equiv 2 \pmod{3},\;\; A + B + C \equiv 0 \pmod{3} \;\Rightarrow\; C \equiv 1 \pmod{3}$$

💡 Subtracting one constraint from another to isolate a single variable is a Grade 6 "use equations as questions about which values work" move — exactly the substitution idea.

#3 Eliminate Possibilities 4.OA.B.4 Step 4

Apply the rule "$C$ leaves remainder $1$ when divided by $3$" to the five answer choices.
Compute each choice mod $3$ and keep the one whose remainder is $1$.

$1 \div 3 \to r = 1$,\; $2 \div 3 \to r = 2$,\; $3 \div 3 \to r = 0$,\; $5 \div 3 \to r = 2$,\; $8 \div 3 \to r = 2$

💡 Checking each candidate's remainder against the required remainder is the Grade 4 "factors and multiples" skill in eliminate-possibilities form.

#3 Eliminate Possibilities 6.EE.B.5 Step 5

Only $C = 1$ leaves remainder $1$ on division by $3$, so that is the answer.

$$C = 1 \;\Rightarrow\; \textbf{(A)}$$

💡 Reading off the lone surviving choice after elimination is the final Grade 6 "which value makes the equation true" step.

[1] #7 4.OA.B.4 Subproblem 1: turn the first number's divisibility into a fact about $A + B$. Ad

[2] #7 4.OA.B.4 Subproblem 2: do the same for the second number. Add the known digits of $326AB4

[3] #7 6.EE.B.5 Combine the two facts. Step 1 says $1 + A + B$ is a multiple of $3$, so $A + B$

[4] #3 4.OA.B.4 Apply the rule "$C$ leaves remainder $1$ when divided by $3$" to the five answer

[5] #3 6.EE.B.5 Only $C = 1$ leaves remainder $1$ on division by $3$, so that is the answer.

Review

Reasonableness: Test the answer with a concrete $A$ and $B$. Pick $A = 1$, $B = 1$, so $A + B = 2$ (leaves remainder $2$ mod $3$, matching Step 1). First number: $7411521$. Digit sum $= 7+4+1+1+5+2+1 = 21 = 3 \times 7$, divisible by $3$. Second number with $C = 1$: $3261141$. Digit sum $= 3+2+6+1+1+4+1 = 18 = 3 \times 6$, divisible by $3$. Both checks pass, so $C = 1$ really works and choice (A) is correct.

Alternative: Tool #6 (Guess and Check) directly on the choices: pick any sample $A,B$ with $A+B \equiv 2 \pmod 3$ — say $A = B = 1$ — and test each choice for $C$ in the second digit sum $15 + 1 + 1 + C = 17 + C$. We need $17 + C$ divisible by $3$, i.e. $C \equiv 1 \pmod 3$. Trying the choices: $17+1=18$ (yes), $17+2=19$ (no), $17+3=20$ (no), $17+5=22$ (no), $17+8=25$ (no). Only (A) $C = 1$ works.

CCSS standards used (min grade 6)

4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite (Applying the divisibility-by-$3$ rule (digit sum is a multiple of $3$) to both $7$-digit numbers and checking which answer choices leave remainder $1$ when divided by $3$.)
6.EE.B.5 Understand solving an equation as a process of answering a question: which values from a specified set make the equation true (Subtracting the first constraint ($A+B \equiv 2 \pmod 3$) from the second ($A+B+C \equiv 0 \pmod 3$) to isolate $C$, and identifying which of the five listed values for $C$ satisfies the resulting condition.)

⭐ This AMC 8 problem only needs the Grade 4 divisibility-by-$3$ trick (add the digits) plus a Grade 6 "which value fits both rules?" check that you already know!

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: A

Review

CCSS standards used (min grade 6)

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